Constructing a Weighted Random Graph

I want to create a weighted random graph (in contrast to the unweighted Erdős–Rényi model). I have a list of weights (derived from a real-world network, very skewed distribution that most weights are 1 and the rest of them are between 6-8). My plan is to first create an Erdős–Rényi model and randomly reassign the weights according to my list of weights. Does it sound like the right approach or my new network essentially lost the key characteristics of a random graph?